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Transversals in uniform hypergraphs with property (7,2)

Dmitry G. Fon-Der-Flaass, Alexandr V. Kostochka, Douglas R. Woodall
1999 Discrete Mathematics
Let f(r; p; t) (p ¿ t¿1, r¿2) be the maximum of the cardinality of a minimum transversal over all r-uniform hypergraphs H possessing the property that every subhypergraph of H with p edges has a transversal  ...  The values of f(r; p; 2) for p =3; 4; 5; 6 were found in Erdős et al. (Siberian Adv. Math. 2 (1992) 82-88). We give bounds on f(r; 7; 2), partially answering a question in Erdős et al. (1992) .  ...  For a hypergraph H = (V; E), a transversal is a transversal of E. Say that B possesses the property (p; t) if (F)6t for every F ⊂ B with |F| = p.  ...

P-apex graphs

Mieczysław Borowiecki, Ewa Drgas-Burchardt, Elżbieta Sidorowicz
2018 Discussiones Mathematicae Graph Theory
We give a sharp upper bound on the number of vertices of graphs in C(P(1)) and we give a construction of graphs in C(P(k)) of relatively large order for k ≥ 2.  ...  In order to obtain desired results we exploit some hypergraph tools and this technique gives a new result in the hypergraph theory.  ...  In this section we prove that an r-uniform τ -vertex 2-critical hypergraph has at most (r+2) 2 4 vertices.  ...

Eckhoff's Problem on Convex Sets in the Plane

Adam S. Jobson, André E. Kézdy, Jenő Lehel
2021 Electronic Journal of Combinatorics
The case \$m=2\$ was verified by Nadler and by Perles. Here we show that Eckhoff 's conjecture follows from an old conjecture due to Szemerédi and Petruska concerning \$3\$-uniform hypergraphs.  ...  This conjecture is still open in general; its solution for a few special cases answers Eckhoff's problem for \$m=3,4\$. A new proof for the case \$m=2\$ is also presented.  ...  (a)] states that m+2 2 is the maximum order of a 3-uniform τ -critical hypergraph with transversal number m.  ...

Upper bound on the order of τ-critical hypergraphs

A Gyárfás, J Lehel, Zs Tuza
1982 Journal of combinatorial theory. Series B (Print)
The right order of magnitude for the maximal number of vertices in an r-uniform r-critical hypergraph H is achieved by obtaining an upper bound of O(r(H)'-').  ...  Also, T' nf # 0, Proof: Let H be an r-uniform r-critical hypergraph with r(H) = t. We proceed in five steps. Step 1. Let T be the t-uniform hypergraph whose edges are the f-element transversals of H.  ...  It is clear, however, that for every r > 2 and 1 < k < t, m(r, t, k, 0) is equal to the maximal order of r-uniform hypergraphs with transversal number t and critical in the stronger sense expressed by  ...

Page 7224 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews
An r-uniform hypergraph of order p without (r + 1)-cliques is called a (p,r)-hypergraph.  ...  Here we investigate uniform hypergraphs with an analogous be- haviour of their entropy. The main result is the characterization of 3-uniform hypergraphs having this entropy splitting property.  ...

Sprague-Grundy Function of Matroids and Related Hypergraphs [article]

Endre Boros, Vladimir Gurvich, Nhan Bao Ho, Kazuhisa Makino, Peter Mursic
2019 arXiv   pre-print
In particular we characterize all 2-uniform hypergraphs (that is graphs) and all matroids for which the formula works. We show that all self-dual matroids are included in this class.  ...  In this paper we give an explicit formula that describes the Sprague-Grundy function of hypergraph NIM for several classes of hypergraphs.  ...  (i) A JM hypergraph is minimal transversal-free. (ii) A graph (that is, a 2-uniform hypergraph) is JM if and only if it is connected and minimal transversal-free.  ...

Decomposing 1-Sperner hypergraphs [article]

Endre Boros, Vladimir Gurvich, Martin Milanič
2018 arXiv   pre-print
In particular, we obtain bounds on the size of 1-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and  ...  We introduce in this paper the class of 1-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one.  ...  The work of the third author is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285, research projects N1-0032, J1-6720, and J1-7051).  ...

Decomposing 1-Sperner Hypergraphs

Endre Boros, Vladimir Gurvich, Martin Milanič
2019 Electronic Journal of Combinatorics
In particular, we obtain bounds on the size of \$1\$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals,  ...  We introduce in this paper the class of \$1\$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one.  ...  The work for this paper was done in the framework of bilateral projects between Slovenia and the USA, partially financed by the Slovenian Research Agency (BI-US/14-15-050, BI-US/16-17-030, and BI-US/18  ...

Polynomial Time SAT Decision, Hypergraph Transversals and the Hermitian Rank [chapter]

Nicola Galesi, Oliver Kullmann
2005 Lecture Notes in Computer Science
We show polynomial time SAT decision of the class of formulas with hermitian rank at most one, applying methods from hypergraph transversal theory.  ...  Combining graph theory and linear algebra, we study SAT problems of low "linear algebra complexity", considering formulas with bounded hermitian rank.  ...  challenges in hard combinatorial problems and in optimization under 'uncertainty'".  ...

Bounding the Number of Minimal Transversals in Tripartite 3-Uniform Hypergraphs [article]

Alexandre Bazin
2021 arXiv   pre-print
We bound the number of minimal hypergraph transversals that arise in tri-partite 3-uniform hypergraphs, a class commonly found in applications dealing with data.  ...  Let H be such a hypergraph on a set of vertices V. We give a lower bound of 1.4977 |V | and an upper bound of 1.5012 |V | .  ...  When all the hyperedges of a hypergraph have the same arity p, we call it a p-uniform hypergraph.  ...

Maximum degree and fractional matchings in uniform hypergraphs

Zoltán Füredi
1981 Combinatorica
I would like to express my thanks to P. Frankl and I. Bárány for their help.  ...  For the projective If tr is D-regular and (2) FRACTIoNAL MATCHINGS IN HYPERGRÁ'PHS r57 p|ane 3, with r>3, J.  ...  Applying the induction hypothesis to #r with parameters y-2 and p, and using that upper bound for ltr'zl which follows from (8), we get | # | : I trl + lr rl = (v -2) (rzr) * p i-2 (rzr) + Z -(rr).  ...

Transversal numbers of uniform hypergraphs

Noga Alon
1990 Graphs and Combinatorics
For k ~ 1 define ck = supz(H)/(ra + n), where H ranges over all k-uniform hypergraphs with n vertices and m edges. Applying probabilistic arguments we show that ck = (1 + o ( 1 ) )~, r.  ...  The transversal number ~(H) of a hypergraph H is the minimum eardinality of a set of vertices that intersects all edges of H.  ...  Let H = (V, E) be a k- Hypergraphs with Relatively Large Transversal Numbers In this section we assume, whenever it is needed, that k is sufficiently large.  ...

Minimum number of elements representing a set system of given rank

Zsolt Tuza
1989 Journal of combinatorial theory. Series A
Moreover, we determine the maximum cardinality of strongly independent vertex sets in r-critical and intersecting v-critical hypergraphs of given rank, and describe the extremal structures.  ...  INTRODUCTION As defined in [ 11, the transversal number r(Z) of a hypergraph (set system) L%?  ...  Then let & be the r-uniform hypergraph whose edge set E consists of the r-tuples of form Pu {up} and P' u {up}.  ...

Uniquely \$K_r^{(k)}\$-Saturated Hypergraphs

András Gyárfás, Stephen G. Hartke, Charles Viss
2018 Electronic Journal of Combinatorics
This is in contrast to the case \$k=2\$ and \$r=3\$ where only the Moore graphs of diameter two have this property.  ...  For integers \$k,r,n\$ such that \$2\leqslant k <r<n\$, a \$k\$-uniform hypergraph \$H\$ with \$n\$ vertices is uniquely \$K_r^{(k)}\$-saturated if \$H\$ does not contain \$K_r^{(k)}\$ but adding to \$H\$ any \$k\$-set that  ...  On the other hand, in Tables 2 through 5 , we see that many primitive uniquely K (k) rsaturated hypergraphs exist with uniformity at least 3.  ...